I) TRICKY ESR FOR THE SQUARE ROOT OF 13

Q: What is the square root of 13?

..__
√13    = ?  

Okay, let's apply ESR in taking the square root of 13... 

.._3._                                We choose 3 because 13 is in-between 16 (the square of 4)
√13.00'00'00'00                  and 9 (square of 3) 
________________

4.0²  = 16.00                        Do the Square Averaging Method
9/8<<
           14.---
7/6<<
3.5² =  12.25  .
           28.---   / 2
           14.---
_________________

3.7²   =  9.49
3x14  =  4 2    .
            (Blank) OVER


3.6²        =     9.36
3x12       =     3 6    .
                 < 12.96 >
____________________

.._3. 6                      (Write down 6 after 3...)        
√13.00'00'00'00        

___________________

3.6²           =     9.36
3x12          =     3 6    .
                   < 12.96 >
                          36    .
3.65²        =    13.32       <<< Middle Square
4/3<<
                       13.---        <<< Lower  Quarter Square
2/1<<  
3.6²          =    12.96    .  
                       26.---  ./ 2
                       13.---
_________________

3.62²   =  12.96'04
36x4  =         14'4    .
                    (Blank)  OVER


3.61²   =  12.96'01               <<< Later, you'll need to compute this...
36x2  =          7'2  .
                   (Blank)  OVER

3.60²  = < 12.96'00 > 

 Take Note:  
Sampling the digits 2 and 1 give us a sum values  which are 'greater than' 13. In this case, we must  resort into choosing  the digit "0" as the next digit after 3.6. don't forget to paste '00' after 12.96 
_________________

.._3. 6   0                 .    (Write down 0 after 3.6 ...)        
√13.00'00'00'00  

_________________

3.60²      = < 12.96'00 > 
                          3 60   .
3.605²   =      12.99'60
7/6 <<<
                     13.---
9/8 <<<
3.61²      =     13.0321  . <<< Solve the sampling for 3.61  (See Above, Left Side))
                     26.---   / 2
                     13.---
_________________

3.607²   =  12.96'00'49
" x 14   =          5 04 0   .
                      (Blank)    OVER


3.606²   =  12.96'00'36           <<< Later, you'll also need to compute this...
" x 12  =           4'32'0  .
                      (Blank)    OVER


 3.605²   =    < 12.99'60'25 > 

Take Note:
Again, the two digits 7 and 6 failed to fit as the next candidate, so we must choose the digit '5" as the required digit. Simply copy the value for the middle square  "3.605²  = 12.99'60" but DON'T FORGET TO PASTE '25', so you'll not go wrong in computing for the next digit we're looking for. 

____________________

.._3. 6   0   5           (Write down 5 after 3.60 ...)        
√13.00'00'00'00  
___________________

3.605²     =   12.99'60'25
                            36'05  .
3.6055²    =   12.99'96'30
7/6 <<<
                     13.---
9/8 <<<
3.601      =     13.00'32'36  .
                      26.---     / 2
                      13.---
____________________

3.6057²    =  12.99'60'25'49
" x 14      =           50'47'0    .
                            (Blank)   OVER


3.6056²    =  12.99'60'25'36
" x 12     =            43'26'0    .
                            (Blank)   OVER

3.6055²   = < 12.99'96'30'25 >   <<< Don't forget to paste '25'
______________________

.._3. 6   0   5    5     (Finally, write down 5 after 3.605 ...)        
√13.00'00'00'00